Chapter 12.8, Problem 28HP

To determine

To find the ratio of the volume of cylinder C to the volume of cylinder D.

Answer to Problem 28HP

  $8:135$

Explanation of Solution

Given:

  $\frac{\text{Volume of cylinder A}}{\text{Volume of cylinder B}}=\frac{1}{5}$

Cylinder A is similar to cylinder C with a scale factor of $1:2$ .

Cylinder B is similar to cylinder D with a scale factor of $1:3$ .

Concept used:

If the two cylinders are similar, then the ratio of the volumes of the cylinders equals the cube of their scale factor.

Calculation:

  $\frac{\text{Volume of cylinder A}}{\text{Volume of cylinder B}}=\frac{1}{5}$

Since cylinder A is similar to cylinder C with a scale factor of $1:2$ ,

  $\frac{\text{Volume of cylinder A}}{\text{Volume of cylinder C}}={\left(\frac{1}{2}\right)}^3$

  $=\frac{1}{8}$

Since cylinder B is similar to cylinder D with a scale factor of $1:3$ ,

  $\frac{\text{Volume of cylinder B}}{\text{Volume of cylinder D}}={\left(\frac{1}{3}\right)}^3$

  $=\frac{1}{27}$

Now, using these ratios to get the ratio of C to D.

If $\frac{\text{Volume of cylinder A}}{\text{Volume of cylinder B}}=\frac{1}{5}$ and $\frac{\text{Volume of cylinder A}}{\text{Volume of cylinder C}}=\frac{1}{8}$

Then.

  $\begin{array}{l}\Rightarrow \frac{\text{Volume of cylinder B}}{\text{Volume of cylinder C}}\\ \Rightarrow \frac{\frac{\text{Volume of cylinder B}}{\text{Volume of cylinder A}}}{\frac{\text{Volume of cylinder C}}{\text{Volume of cylinder A}}}\\ \Rightarrow \frac{5}{8}\\ \Rightarrow \frac{1}{\frac{8}{5}}\end{array}$

If $\frac{\text{Volume of cylinder B}}{\text{Volume of cylinder C}}=\frac{1}{\frac{8}{5}}$ and $\frac{\text{Volume of cylinder B}}{\text{Volume of cylinder D}}=\frac{1}{27}$

Then.

  $\begin{array}{l}\Rightarrow \frac{\text{Volume of cylinder C}}{\text{Volume of cylinder D}}\\ \Rightarrow \frac{\frac{\text{Volume of cylinder C}}{\text{Volume of cylinder B}}}{\frac{\text{Volume of cylinder D}}{\text{Volume of cylinder B}}}\\ \Rightarrow \frac{\frac{8}{5}}{27}\\ \Rightarrow \frac{8}{135}\end{array}$

Conclusion:

Therefore, the ratio of volume of cylinder C to the volume of cylinder D is $8:135$ .